Rub each duck inside and out with salt (¼ cup for each). Place both on a rack on a baking sheet and refrigerate, uncovered, 24 hours. The next day, rinse off all of the salt; set the ducks aside. In a pot large enough to hold one duck, combine the sugar and water and bring to a boil. Add one duck and poach at a rolling boil for 12 minutes (use a Pyrex bowl, weighted with water or a can, to keep the duck submerged if necessary). Remove the duck with 2 large forks to drain on an ovenproof wire rack for 30 minutes. Repeat to poach and drain the second duck.

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Preheat the oven to 400°F. Set the wire rack (with ducks) on a baking sheet. Roast the ducks, breast side up, until the skin has completely blackened, about 1½ hours. Turn and roast until the bottom is blackened and the drumsticks move freely, about 30 more minutes.

Meanwhile, to make the sauce, combine the sugar and water in a small, heavy-bottomed saucepan. Cook over medium-high heat without stirring, shaking the pan from time to time, until the water has evaporated and the sugar has caramelized to a light-brown color, about 10 minutes. Remove from heat, slowly add the vinegar (stand back, the caramel will sputter), and stir; return to the heat and simmer 5 minutes. Add the juice and cook 2 minutes. Add the espresso and the orange and coffee liqueurs. Turn heat to high and boil until thickened enough to coat the back of a spoon, 20 to 30 minutes.

To serve, cut each duck into 8 pieces with bone (separate leg quarters from breast and cut thigh from drumstick; cut breasts in half) and accompany with the sauce. Serves 6.

Bring a 3-quart pot of salted water to a boil over high heat. Add the celery root, return to the boil, and boil until tender, about 6 minutes. With a slotted spoon, transfer to a large baking sheet. Add the turnips to the pot, return to a boil, and cook until tender, about 5 minutes. Transfer to the baking sheet. Add the pears, return to a boil, and cook 2 minutes; transfer to the baking sheet. Add the beets, return to a boil, and simmer 20 to 25 minutes. Drain the beets and transfer to a large plate. Let cool slightly, then peel and quarter; set aside on the plate with the rest of the fruit and vegetables.

Return the pot to the stove. Add the olive oil and heat over medium heat. Add the garlic, shallot and thyme and cook until softened, about 5 minutes. Add the stock, bring to a boil over high heat, and boil until reduced to 1 cup, about 5 minutes. Add the cooked celery root, turnips and pears. Reduce the heat to medium-high, cover and cook, folding gently a few times with a heatproof rubber spatula, until vegetables are heated through, 3 to 5 minutes. Add the beets and chestnuts and season with ½ teaspoon salt and ½ teaspoon pepper. Cover and cook over medium heat until heated through, about 3 minutes. Gently stir in the butter and transfer to a bowl. Serves 6.